Chains Of Tangent Hard Spheres Research Articles

Primitive models of chemical association. II. Polymerization into flexible chain molecules of prescribed length

The structural properties of the totally flexible sticky two-point (S2P) model for polymerization into chain molecules of fixed length are studied. The model is represented by an n-component mixture of hard spheres of the same size with species 2,…,n−1 bearing two attractive sticky sites A and B, randomly distributed on the surface. The hard spheres of species 1 and n have only one site per particle, site B for species 1 and site A for species n. Due to the specific choice for the attractive interaction, which is present only between site B of the particles of species a and site A of the particles of species a+1, this version of the S2P model represents an associating fluid that polymerizes into freely jointed tangent hard-sphere chain molecules. The correlation functions of this model are studied at all degrees of association using a recently obtained general solution of the polymer Percus–Yevick (PPY) approximation [Yu. Kalyuzhnyi and P. Cummings, J. Chem. Phys. 103, 3265 (1995)]. Comparison of the results of the present theory in the complete association limit with corresponding computer-simulation results and results of other theories is presented and discussed. The complete-association results constitute a quantitatively successful theory of the mean monomer–monomer distribution functions for n⩽16 but for n=50 these functions are no longer quantitatively accurate.

Simulation of free energy without particle insertion in the NPT ensemble

A temperature perturbation method for a direct Monte Carlo simulation of the residual free energy is extended to the isothermal–isobaric ensemble. Unlike many other free energy or phase equilibrium methods, particle insertion is avoided. At the same time, the usual Boltzmann sampling is maintained. The results show excellent agreement with equations of state and thermodynamic integration for fluids of hard dumbbells and linear chains of tangent hard spheres. The results cover states at which insertion techniques fail.

Computer simulation study of the approximations associated with the generalized Flory theories

The chain increment method and configurational bias Monte Carlo methods are used to test the approximations made in the derivation of the generalized Flory-Dimer (GF-D) theory for tangent hard sphere chains. Insertion probabilities and residual chemical potentials are calculated for hard chain fluids containing chains of length n=4, 8, 16, and 32 at monomer densities, ρM, up to 0.8. We find that the largest errors in the GF-D theory are those associated with assuming that the probability of inserting a monomer into a chain fluid is approximately equal to the probability of inserting a monomer into a monomer fluid, as predicted by the Carnahan–Starling equation of state. The errors in the incremental compressibility factor of the second segment associated with assuming that the conditional probability of inserting a second bead next to the first bead in a chain fluid is approximately equal to the probability of inserting a second bead next to the first bead in a dimer fluid as predicted by combining the Carnahan–Starling and Tildesley–Streett equations of state are relatively small. Consistent with the findings of Mooij and Frenkel, we find that these two approximations lead to an overprediction of the incremental contributions to the compressibility factor. Despite the overprediction of the incremental contributions to the compressibility factor of the first segment, the GF-D equation of state accurately predicts the compressibility of hard chains; this accuracy is traced to (1) the insensitivity of the compressibility factor to errors in the insertion probability and (2) cancellation of errors in the incremental compressibility factor of the first segment with small cumulative errors in the incremental compressibility factors of the third and subsequent segments.

Molecular dynamics study of entangled hard-chain fluids

By applying efficient computational algorithms to the simplest off-lattice polymer model–the freely-jointed tangent hard-sphere chain–we have been able to perform molecular dynamics simulations long enough to probe chain dynamics in the entangled regime. Chain lengths range from 8 to 192 segments while volume fractions range from 0.3 to 0.45. Analysis of the mean-square displacement (MSD), Rouse modes, scattering functions, and end-to-end vector correlations provides information about chain motion. Chain dynamics are compared with predictions of the Rouse model for short chains and the tube model of Doi and Edwards for long chains. The mean-square displacement for the inner segments of the longest chains are consistent with predictions of the tube model, reproducing the three scaling regimes that are postulated to occur. In addition, anomalous diffusive behavior in the atomic MSD of the inner segments is observed at long times as the inner segments cross over into the free diffusion limit. Rouse-mode autocorrelation functions decay non-exponentially and do not exhibit scaling consistent with the tube model. Definitive plateau-like behaviors are observed in the density–density correlations, normal coordinate decay, and end-to-end vector relaxation of the 192-mer fluids at the highest density.

Collapse of a polymer chain: A Born–Green–Yvon integral equation study

A Born–Green–Yvon (BGY) type integral equation is developed for the intramolecular distribution functions of an isolated flexible polymer chain. The polymer is modeled as a linear array of n identical spherical interaction sites connected by universal joints of bond length σ. In particular we study chains composed of up to n=400 square-well spheres with hard-core diameters σ and well diameters λσ (1≤λ≤2). Intramolecular distribution functions and the resulting average configurational and energetic properties are computed over a wide range of temperatures. In the high temperature (good solvent) limit this model is identical to the tangent hard-sphere chain. With decreasing temperature (worsening solvent) the square-well chain undergoes a collapse transition identified by a sudden reduction in chain dimensions and a peak in the single chain specific heat. Extensive comparison is made between the BGY results and Monte Carlo results for square-well chains with λ=1.5. The BGY theory is extremely accurate for square-well 4-mers at all temperatures. For longer chains the theory yields reasonably accurate results for reduced temperatures greater than T*≊1 (expanded and theta states) and qualitatively correct behavior for T*<1 (collapsed state). Very accurate values for the theta temperatures for square-well chains with 1.25≤λ≤2.0 are also obtained.

The theoretical prediction of the cricital points of alkanes, perfluoroalkanes, and their mixtures using bonded hard-sphere (BHS) theory

The adequacy of the recently developed bonded hard-sphere (BHS) theory in describing the critical behavior of the homologous series of the alkanes and perfluoroalkanes is examined in this work. A simple united atom model, formed from chains of tangent hard spheres, reproduces the major experimental trends and provides good quantitative agreement for systems with two or more carbon atoms. This simple model cannot, however, reproduce the anomalous behavior of the critical pressure of the alkane series: the values of the critical pressure and temperature for methane are smaller than expected. A more sophisticated distributed-site model, which takes explicit account of the backbone and substituent atoms, reproduces this anomalous behavior. The BHS theory has also been used to predict the upper critical solution temperatures of alkane + perfluoroalkane mixtures. For most systems, the segment-segment parameters are fitted to the butane + perfluorobutane system, although in the case of mixtures containing methane, methane + perfluoromethane parameters must be used. Excellent qualitative agreement with experimental data is seen. This indicates the strength of the BHS approach as a type of group contribution method.

Linear dependence on chain length for the thermodynamic properties of tangent hard-sphere chains

Equilibrium molecular dynamics simulation techniques are used to obtain accurate compressibility factors (⩽0·1% error) for tangent hard-sphere (THS) chains of lengths 2–8, 16, 32, 64, 96, and 192. Our simulation results show that, within simulation statistical errors, the dependence of compressibility factors on chain length approaches linearity very rapidly. At volume fractions of 0·4 or above the linearity starts at a chain length of 3, while at volume fractions of 0·1 or above the linearity starts at a chain length of 6. The thermodynamic perturbation theory (TPT) and the generalized Flory (GF) theory equations of state are extended to become a linear combination of the compressibility factors of any two reference THS chain fluids. It is found that extended GF theory is identical to extended TPT when the excluded volumes of chains and reference chains are assumed to be linearly dependent on the chain length. Our simulation data implies that a near exact THS chain equation of state can be obtained from ...

Excluded volume for a pair of linear chains of tangent hard spheres with an arbitrary relative orientation

Chain molecules formed from m tangent hard-sphere segments are of particular interest in studies of liquid crystalline phases because they represent one of the simplest models which can account for molecular flexibility. In this paper we obtain analytical expressions for the excluded volume for a pair of linear homonuclear chains with an arbitrary relative orientation, and the corresponding expressions for the second virial coefficient. A commonly used molecular theory of anisotropic phases such as the nematic is the approach of Onsager; a simple analytical expression for the orientationally dependent excluded volume for a pair of molecules is required for the efficient minimisation of the Onsager Helmholtz free energy functional at the level of the second virial coefficient. Approximate expressions are examined for the excluded volume, together with a simple analytical function which provides a very accurate description of the exact results. A simple exact relation is obtained for the second virial coefficient of the isotropic system for arbitrary chain lengths using the exact result of Isihara for the dimer, i.e., for m = 2. Recent Monte Carlo numerical integration data for chains of m = 3 to 7 segments are in agreement with exact results. In a future contribution, we hope to compare the results of simulations for the isotropic-nematic phase transition exhibited by linear chains of tangent hard spheres with the Onsager predictions using these expressions for the excluded volume.

Large-Scale Molecular Dynamics Study of Entangled Hard-Chain Fluids

Equilibrium molecular dynamics is used to simulate fluids comprised of chains of tangent hard spheres. Reptation theory predictions of segmental motion are compared with simulation results. In addition to the usual tube confinement, a second entanglement effect is observed. As the chain disengages from the tube, persistent interchain contacts cause a plateau in the segment mean-square displacement and subsequent accelerated diffusion. Associated with the plateau in the mean-square displacement is a corresponding delay in relaxation of the end-to-end vector as interior chain segments are extended during disentanglement.

Chemical potential and equations of state of hard core chain molecules

A novel approach is presented for the development of equations of state for chain molecules. The basic assumptions of the approach are supported by results of computer simulations of the chemical potential of athermal chains. Our model establishes a bridge between some elements of Wertheim’s thermodynamic perturbation theory and the generalized Flory theory. New equations of state are presented for freely jointed tangent hard-sphere chains which are shown to be more accurate than other existing equations. Extensions of our approach are also presented for branched polymers and blends of polymers.

Variational theory for Lennard-Jones chains

The Helmholtz energy and pressure for Lennard-Jones chains are derived using a variational first-order perturbation theory. The reference system, comprising freely jointed tangent hard-sphere chains is solved in the Percus-Yevick approximation. The optimal diameter of segments in the hard-sphere chain reference is determined by minimizing the free energy through the Gibbs-Bogoliubov inequality. Unlike previous perturbation theories for non-aggregated Lennard-Jones spheres, the resulting hard-sphere segment diameter depends on chain length in addition to density and temperature. The resulting theory exhibits excellent agreement with MC simulation, over a wide range of conditions, without adjustable parameters.

Fused hard-sphere chain molecules: Comparison between Monte Carlo simulation for the bulk pressure and generalized Flory theories

We report Monte Carlo simulation results for the bulk pressure of fused-hard-sphere (FHS) chain fluids with bond-length-to-bead-diameter ratios ≊ 0.4 at chain lengths n=4, 8 and 16. We also report density profiles for FHS chain fluids at a hard wall. The results for the compressibility factor are compared to results from extensions of the Generalized Flory (GF) and Generalized Flory Dimer (GFD) theories proposed by Yethiraj et al. and by us. Our new GF theory, GF-AB, significantly improves the prediction of the bulk pressure of fused-hard-sphere chains over the GFD theories proposed by Yethiraj et al. and by us although the GFD theories give slightly better low-density results. The GFD-A theory, the GFD-B theory and the new theories (GF-AB, GFD-AB, and GFD-AC) satisfy the exact zero-bonding-length limit. All theories considered recover the GF or GFD theories at the tangent hard-sphere chain limit.

Polymer Born–Green–Yvon equation with proper triplet superposition approximation. Results for hard-sphere chains

A site–site Born–Green–Yvon (BGY) equation is derived for polymeric fluids. This relates the pair and triplet site distribution functions, and superposition approximations for the latter are analyzed. It is shown that the pair functions to be superposed are uniquely determined by the exact normalization equations and asymptotic conditions. The Kirkwood superposition of pair distribution functions is shown to be valid only for the case of sites on three different polymers; for the cases of two or three sites on the same polymer different pair functions must be superposed. The polymer BGY equation is derived for a soft bonding potential between adjacent sites; the result for infinitely stiff bonds is given as a limiting case. Numerical results are obtained for soft and stiff tangent hard-sphere chains, and comparison is made with simulations for packing fractions up to 0.4 and chains with up to 12 sites.

Molecular dynamics study of transport coefficients for hard-chain fluids

Equilibrium molecular dynamics is used to simulate fluids containing molecules modeled as chains of tangent hard spheres. A partially vectorized, efficient algorithm based on the Rapaport method has been designed that allows for very long simulation times and permits calculation of transport coefficients for short chain fluids at liquid-like densities. The self-diffusion coefficient, shear and longitudinal viscosities, and thermal conductivity are calculated for chains of length 2, 4, 8, and 16 at volume fractions ranging from 0.1 to 0.5 using a mean-square displacement approach. Results from the velocity autocorrelation functions provide information about chain motion in the bulk phase. Transport properties for the hard-sphere fluid have also been calculated for systems of 512 particles. Results for chain fluids are compared to results for hard-spheres and to the corresponding Enskog theory.

Monte Carlo Simulation of Off-Lattice Polymer Chains: Effective Pair Potentials in Dilute Solution

Monte Carlo methods are used to calculate the mean squared radius of gyration, , potential of mean force, U(r), and second osmotic virial coefficient, B 2 , of polymer chains. Two model of polymers as off-lattice chains of tangent hard spheres are used. In the first, the beads of the chains are simply hard spheres (the athermal case). In the second, the beads of the chains are allowed to interact with nonadjacent beads on the same chain and with beads on a different chain via a square-well potential. In both model, conformations of chains with 50, 100, 200, and 500 segments were generated. In the square-well model, square-well potentials e=0.15kT, -0.3kT, and -0.45kT were used

Self-consistent integral equation theory for the equilibrium properties of polymer solutions

Self-consistent RISM integral equation theory of polymeric liquids is used to investigate the equilibrium properties of polymer solutions and melts. Density functional formalism is employed to derive the effective medium-induced potential between two sites on a polymer chain. The resulting expression leads to a coupling between the interchain and intrachain pair correlation functions. An approximate method for the calculation of the effective intrachain pair correlations is examined. The method employs Monte Carlo computer simulations of a single polymer chain interacting via local chemical interactions, long-range excluded volume, and the self-consistently determined medium-induced potential in the condensed phase to compute the intrachain pair correlations which are consistent with the interchain packing. Results for solutions of tangent hard-sphere chains are compared to recent off-lattice Monte Carlo simulations for various chain lengths and densities. In particular, it was found that the theory slightly overestimates the average size of the polymers in solution, while the density dependence of the average size of the chains is in good agreement with the Monte Carlo simulations. The inclusion of the self-consistent determination of the intrachain correlations is found to significantly alter the interchain pair correlation function at low and intermediate densities.

Equations of state for square well chain fluids using the generalized flory approach

Abstract The application of the Generalized Flory-Dimer theory (GFD) to square well chain fluids is described. Molecules are modelled as chains of tangent hard spheres that interact via a site-site square well potential u(rij), where sites i and j are on different chains. The derivation of the GFD theory is reviewed; this is based on a combination of mean-field and geometric arguments. The resulting equation of state is in very good agreement with Monte Carlo computer simulation results for chains of length n = 16 and 32 at reduced temperatures T∗ = 2.0 and 3.0 and volume fractions η = 0.1 to 0.4.

Intermolecular site–site correlation functions of athermal hard-sphere chains: Analytic integral equation theory

The intermolecular site–site correlation function gij(r) of freely jointed athermal tangent hard-sphere chains (HSC) are determined based on a particle–particle description of the chain system. We developed a mathematical framework through which the functions gij(r) can be obtained within the context of the Percus–Yevick (PY) integral equation theory. The computational scheme involves the analytic solution to the structure of a multicomponent particle mixture that is subject to some specified connectivity constraints. In particular, we determined the site–site correlation functions gij(r) of homonuclear 4-mer chains, and homonuclear 4-mer chains in a hard-sphere solvent. Analytic expressions for the site–site and site–solvent correlation functions at contact, i.e., gij(σ+), are obtained. Site–site correlation functions determined from the PY theory show the ‘‘correlation hole’’ behavior at low density. Comparison of the PY results with computer simulation data shows that the theory yields accurate values for an averaged ‘‘total’’ correlation function, and captures the essential features of the site–site correlation functions gij(r). In addition the effect of hard-sphere solvent particles on the structure of HSC molecules is also investigated.

Percus-Yevick integral-equation theory for athermal hard-sphere chains

A theoretical method for the modelling of athermal freely jointed tangent hard-sphere chain fluids, of fixed length r, is developed based on a ‘particle-particle’ description of the chain system. This approach is based on the Percus-Yevick (PY) theory in the context of the particle-particle Ornstein-Zernike integral equation subject to some imposed connectivity constraints. Analytical expressions for the compressibility equations of state are derived for homonuclear chains, heteronuclear chains, blends or mixtures of homonuclear and heteronuclear chains, and homonuclear chains in a hard-sphere solvent. The PY compressibility equation of state for the athermal hard-sphere chain system is found to consist of (i) a non-bonded hard-sphere PY compressibility pressure contribution, and (ii) a PY bonding contribution due to chain formation. In the case of homonuclear chains the Percus-Yevick solution is found to yield excellent agreement with computer-simulation data reported in the literature. By replacing the PY hard-sphere compressibility pressure contribution with the Carnahan-Starling hard-sphere pressure, the accuracy of the PY bonding term for homonuclear chains is identified. We are, however, unable to determine the accuracy of the PY compressibility pressure of heteronuclear chains, chain mixtures and homonuclear chains in a hard-sphere solvent since computer-simulation data for these systems are not available.

Generalized flory theories for predicting properties of fluids containing long chain molecules

Two theories have been developed which predict the equation of state for fluids containing chain-like molecules. The Generalized Flory and Generalized Flory-Dimer theories are derived as generalizations of the well-known Flory lattice theory to continuous space. The theories are applied to fluids and fluid mixtures containing molecules which are modelled as chains of tangent hard spheres. The predicted pressures are in very good agreement with Monte Carlo simulations of pure fluids containing chains of 4, 8 and 16 spheres and with Monte Carlo and molecular dynamics simulations of a mixture of hard chains of length n=8 and n=1.